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Signals and Systems – Relation between Convolution and Correlation
Convolution
The convolution is a mathematical operation for combining two signals to form a third signal. In other words, the convolution is a mathematical way which is used to express the relation between the input and output characteristics of an LTI system.
Mathematically, the convolution of two signals is given by,
$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( t-\tau \right )d\tau =\int_{-\infty }^{\infty }x_{2}\left ( \tau \right )x_{1}\left ( t-\tau \right )d\tau}$$
Correlation
The correlation is defined as the measure of similarity between two signals or functions or waveforms. The correlation is of two types viz. cross-correlation and autocorrelation.
The cross correlation between two complex signals 𝑥1(𝑡) and 𝑥2(𝑡) is given by,
$$\mathrm{R_{12}\left ( \tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( t \right )x^{\ast }_{2}\left ( t-\tau \right )dt =\int_{-\infty }^{\infty }x_{1}\left ( t+\tau \right )x^{*}_{2}\left ( t \right )dt}$$
If 𝑥1(𝑡) and 𝑥2(𝑡) are real signals, then,
$$\mathrm{R_{12}\left ( \tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( t \right )x_{2}\left ( t-\tau \right )dt =\int_{-\infty }^{\infty }x_{1}\left ( t+\tau \right )x_{2}\left ( t \right )dt}$$
Relation between Convolution and Correlation
The convolution and correlation are closely related. In order to obtain the crosscorrelation of two real signals 𝑥1(𝑡) and 𝑥2(𝑡), we multiply the signal 𝑥1(𝑡) with function 𝑥2(𝑡) displaced by τ units. Then, the area under the product curve is the cross correlation between the signals 𝑥1(𝑡) and 𝑥2(𝑡) at 𝑡 = 𝜏.
On the other hand, the convolution of signals 𝑥1(𝑡) and 𝑥2(𝑡) at 𝑡 = 𝜏 is obtained by folding the function 𝑥2(𝑡) backward about the vertical axis at the origin [i.e., 𝑥2(−𝑡)] and then multiplied. The area under the product curve is the convolution of the signals 𝑥1(𝑡) and 𝑥2(𝑡) at 𝑡 = 𝜏.
Therefore, it follows that the correlation of signals 𝑥1(𝑡) and 𝑥2(𝑡) is the same as the convolution of signals 𝑥1(𝑡) and 𝑥2(−𝑡).
Analytical Explanation
The resemblance between the convolution and the correlation can be proved analytically as follows −
The convolution of two signals 𝑥1(𝑡) and 𝑥2(−𝑡) is given by,
$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( \tau-t \right )d\tau\: \: \: \cdot \cdot \cdot \left ( 1 \right )}$$
By replacing the variable 𝜏 by another variable p in the integral of Eqn. (1), we get,
$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )=\int_{-\infty }^{\infty }x_{1}\left ( p \right )x_{2}\left ( p-t \right )dp\: \: \: \cdot \cdot \cdot \left ( 2 \right )}$$
Now, replacing the variable 𝑡 by 𝜏 in Eqn. (2), we have,
$$\mathrm{x_{1}\left ( \tau \right )\ast x_{2}\left ( -\tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( p \right )x_{2}\left ( p-\tau \right )dp=R_{12}\left ( \tau \right )}$$
Therefore, the relation between correlation and convolution of two signals is given by,
$$\mathrm{R_{12}\left ( \tau \right )=x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )|_{t=\tau }}$$
Similarly,
$$\mathrm{R_{21}\left ( \tau \right )=x_{2}\left ( t \right )\ast x_{1}\left ( -t \right )|_{t=\tau }}$$
Hence, it proves that the correlation of signals 𝑥1(𝑡) and 𝑥2(𝑡) is equivalent to the convolution of signals 𝑥1(𝑡) and 𝑥2(−𝑡).
Therefore, all the techniques used to evaluate the convolution of two signals can also be applied to find the correlation of the signals directly. Similarly, all the results obtained for the convolution are applicable to the correlation.
Note – If one of the signals is an even signal, let the signal 𝑥2(𝑡) is an even signal [i.e. 𝑥2(𝑡) = 𝑥2(−𝑡)]. Then, the cross-correlation and the convolution of the two signals are equivalent.
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